A problem about bevel after bool modifier
With your lower object (the cut one) selected, click on the Modifiers Tab in the Properties Panel, and check the order of your modifiers in the stack. I have confirmed that when your bevel modifier is above your boolean modifier in the stack, the result is your picture above. When the boolean modifier is moved above the bevel modifer, the bevel then goes all the way around, as it should. You end up with triangles at the corners of the resulting surface, however. But this may not matter, depending on your project. If you don't want triangles at the corners, consider a different approach: Do the boolean only, and apply it. Then in Edit Mode, select the face that makes up the new surface, and press i to inset it. After initiating the offset, but while the tool is still active, press Ctrl to allow the newly resulting face to be extruded in the direction of the face normal. The end result is that you will now have a bevel that doesn't consist of any triangles. The flip side is that the resulting bevel will only exist for this one face, instead of all faces.